The moon itself does not emit light and can only be illuminated by sunlight. Moreover, only half of the lunar surface is illuminated. The part of this half-moon surface that coincides with the half-moon surface facing Earth can be seen by people on Earth, and its shape is called the phases of the moon.
During a lunar month, the phases of the moon constantly change, which is related to the relative position of the sun and the moon. Because the moon revolves around the Earth, it moves from west to east in the sky relative to the sun, traveling about 12° eastward each day, completing a cycle every synodic month. Thus, when the moon reaches the "new moon" position on the first day of the lunar month, people on Earth face a half-moon surface without sunlight. Moreover, since the sun and the moon are close to each other in the same direction and are in strong sunlight, it is impossible to see the moon. Half a month later, when the moon reaches the "full moon" position, the sun and the moon are about 180° apart, and people on Earth face a fully illuminated half-moon surface, allowing them to see a round moon at night. When the moon reaches the "crescent" or "gibbous" positions, people on Earth face a half-moon surface, with one half illuminated by sunlight and the other half completely dark, resulting in a "half-moon". The waning crescent moon on the 22nd and 23rd days of the lunar month is located 90° west of the sun and reaches its peak at dawn. It rises from the east about six hours before sunrise, which is exactly the scene depicted in the folk song "The Half-moon Climbs Up". However, the waxing crescent moon on the 7th and 8th days of the lunar month is located 90° east of the sun and reaches its peak at dusk, so it is impossible to see it rising at noon. Therefore, "The Half-moon Climbs Up" definitely cannot refer to the waxing crescent moon. Similarly, the crescent moon seen at the beginning of the month is always located in the east of the sun and not far away. It hangs low in the west at dusk and soon sets; the crescent moon can only appear in the evening of the last days of the lunar month, located not far from the west of the sun, and appears in the east when the morning light is faint.